act test study guide for leanning english

ACT Test Study
Guide

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Table of Contents
ACT TEST RESOURCES....................................................................................................................... 4
READING................................................................................................................................................... 5
STRATEGY 1: FLYING OVER THE PASSAGE ............................................................................................ 5
STRATEGY 2: CREATING A TENTATIVE SUMMARY ................................................................................. 6
STRATEGY 3: OPENINGS AND ENDINGS ................................................................................................ 7
STRATEGY 4: USING KITCHEN LOGIC .................................................................................................... 7
STRATEGY 5: GETTING INTO THE AUTHOR’S MIND .............................................................................. 8
STRATEGY 6: EMOTIONAL WORDS ......................................................................................................... 8
STRATEGY 7: FINDING THE KEY WORDS ............................................................................................... 9
STRATEGY 8: MAKING PROPER INFERENCES ....................................................................................... 11
STRATEGY 9: APPLYING IDEAS FOR GENERALIZATIONS ..................................................................... 12
STRATEGY 10: USING CONTEXT CLUES ............................................................................................. 12
STRATEGY 11: BREAKING DOWN PASSAGE ORGANIZATION ............................................................. 13
STRATEGY 12: FIRST WORD ANALYSIS ............................................................................................. 15
STRATEGY 13: UNDERSTANDING THE INTIMIDATION......................................................................... 15
STRATEGY 14: FINDING YOUR OPTIMAL PACE ................................................................................... 16
STRATEGY 15: DON’T BE A PERFECTIONIST........................................................................................ 18
STRATEGY 16: FACTUALLY CORRECT, BUT ACTUALLY WRONG ......................................................... 19
STRATEGY 17: DIFFERENT VIEWPOINTS ............................................................................................. 19
STRATEGY 18: EXTRANEOUS INFORMATION ...................................................................................... 20
MATHEMATICS..................................................................................................................................... 21
NUMBER TYPES ....................................................................................................................................... 21

Integers, Odd and Even Numbers, Prime Numbers, Digits.......................................... 21
ADDITION AND MULTIPLICATION OF ODD AND EVEN NUMBERS ........................................................ 21
PERCENT .................................................................................................................................................. 22
Percent less than 100 ................................................................................................................. 22
Percent Greater than 100 ......................................................................................................... 22
Percent less than 1 ...................................................................................................................... 22
Percent Increase/Decrease....................................................................................................... 23
AVERAGE ................................................................................................................................................. 23
WEIGHTED AVERAGE .............................................................................................................................. 24
Average Speed .............................................................................................................................. 25
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PROPERTIES OF SIGNED NUMBERS ....................................................................................................... 25
FACTORING.............................................................................................................................................. 26
PROBABILITY ........................................................................................................................................... 26
GEOMETRIC FIGURES ............................................................................................................................. 27
GEOMETRIC SKILLS AND CONCEPTS ..................................................................................................... 29
Properties of Parallel Lines ....................................................................................................... 29
Angle Relationships ..................................................................................................................... 30
Side Relationships........................................................................................................................ 31
AREA AND PERIMETER ............................................................................................................................ 33
Rectangles....................................................................................................................................... 33
Circles ............................................................................................................................................... 33
Triangles .......................................................................................................................................... 34
VOLUME ................................................................................................................................................... 34
COORDINATE GEOMETRY ....................................................................................................................... 35
ENGLISH.................................................................................................................................................. 37

STRATEGY 1: APOSTROPHES ................................................................................................................. 37
Possessive Nouns ......................................................................................................................... 37
Possessive Personal Pronouns vs. Contractions ............................................................... 37
STRATEGY 2: COMMA ERRORS ............................................................................................................ 38
STRATEGY 3: PROBLEMS WITH REFERENCES ...................................................................................... 40
STRATEGY 4: PROBLEMS WITH AGREEMENT ....................................................................................... 42
STRATEGY 5: LACK OF PARALLELISM ................................................................................................... 45
STRATEGY 6: MISCELLANEOUS PROBLEMS .......................................................................................... 46
SCIENCE .................................................................................................................................................. 51
STRATEGY 1: UNDERSTANDING CHARTS AND TABLES ....................................................................... 51
STRATEGY 2: WHEN VAGUENESS IS CLEAR ........................................................................................ 52
STRATEGY 3: AVOIDING DEFINITES..................................................................................................... 53
STRATEGY 4: USING COMMON SENSE ................................................................................................. 54
STRATEGY 5: INSTINCTS ARE RIGHT ................................................................................................... 55
STRATEGY 6: MAKING NOTES .............................................................................................................. 56
STRATEGY 7: NO FEAR .......................................................................................................................... 56
STRATEGY 8: LOOKING FOR THE CHANGES ......................................................................................... 57
STRATEGY 9: QUICK CHECKS ............................................................................................................... 57
STRATEGY 10: MAKING SUMMARIES ................................................................................................... 59
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STRATEGY 11: LOOKING FOR MATCHES .............................................................................................. 59
STRATEGY 12: PROFESSIONAL EXPERIMENTS ..................................................................................... 60
STRATEGY 13: INFORMATION IS PROVIDED FOR A REASON .............................................................. 61
STRATEGY 14: WATCHING FOR SIMILAR REWORDING ...................................................................... 61
STRATEGY 15: DON’T GET THROWN OFF BY NEW INFORMATION .................................................... 62
STRATEGY 16: LOOKING FOR KEY WORDS ......................................................................................... 63

STRATEGY 17: NARROWING THE SEARCH ........................................................................................... 63
STRATEGY 18: WHAT ABOUT THE OPPOSITE ...................................................................................... 64
STRATEGY 19: YOU’RE NOT EXPECTED TO BE EINSTEIN .................................................................... 65
STRATEGY 20: IDENTIFYING THE KEY COMPONENT ........................................................................... 66
STRATEGY 21: FULFILLING ALL THE REQUIREMENTS.......................................................................... 67
STRATEGY 22: WHEN IT DOESN’T MAKE SENSE – CHECK IT SYSTEMATICALLY ............................. 67

ACT Test Resources

Financial Aid Facts

Scholarship Help

Study Tips and Information
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Reading
The ACT Reading test measures a test taker’s ability to understand,
analyze and evaluate written passages. The passages will contain
material that will be from a variety of sources and on a number of
different topics.
Each of the passages and statements in the Reading test will be
followed by a series of questions covering the content of the passage
or statement, in which you will have to answer questions, which will
demonstrate how well you understand the passages and are able to
draw conclusions about the material.

Strategy 1: Flying Over the Passage
A topic that is hotly debated among test taking circles is whether or
not you should read the reading passages before you read the
question. One theory is that you can save time if you read the
questions first and then go back and read over the passage. Another
theory is that you should read the passage first and then go into the
questions. Both theories have their own individual merit and due to
the differences in ability and preferences among test takers, one
method may work better than another for you.
Our recommended theory is the flyover. You want to spend some time
on the passage, at a bare minimum so that you have a general idea
about what the questions are going to ask and get your mind into the
proper mindset for the series of questions. However, you don’t want
to waste too much time on reading the passage, because much of the
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detail will be forgotten by the time you get to the questions anyway.
Therefore, you should fly over the passage. You should read it very
quickly for a high-level overview (hence the flyover) understanding of
what is contained in the passage.
In part, this is a compromise between the theories that gains most of
the benefits of each. You won’t waste time on the details and yet will
have a general idea of what the passage is about and what to expect.

Strategy 2: Creating a Tentative Summary
After you’ve finished your flyover of the passage, take a few seconds

and compose a tentative mental summary of what you’ve just read.
Try to sort out the details you picked up on and arrange them into a
loose organizational pattern that describes the passage. Remember
that your goal in the flyover is not to check it off of a test-taking list of
things to do. You want there to be some purpose behind the flyover
and having the definite goal of being able to put together a brief
mental summary will allow you to maintain some focus and gain
benefit from the flyover – as opposed to just skimming it for the sake
of skimming it without actually picking up on anything.
As you begin going through the questions and answer choices, if you
get good enough at putting together your mental summaries from
practice, you should be able to eliminate a number of answer choices
that are immediately contrary to your summary. Note, however that if
you find yourself without any good answer choices remaining (because
you’ve eliminated them all) you obviously had to have eliminated the
right answer choice. Don’t hesitate to reopen an answer choice that
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you’ve already “eliminated” from consideration and reconsider it as a
possibility. If you think an answer choice contradicts your initial
summary, you’re probably right, but are not infallible.

Strategy 3: Openings and Endings
A main focus of this flyover will be the opening and ending sentences
in each paragraph. These are likely to contain the main ideas of the
paragraphs and should be mentally tagged for future reference. Try to
remember a vague idea of what the different paragraphs are about,

because this will save you time when answering questions later.
For the most part, make sure you never try to just answer the
questions from this first flyover. Always try to go back and confirm
the answer, as your memory will play tricks on you and the writers of
the test questions may deliberately have planted a trap for you –
remember that they don’t exactly have your best interests at heart.

Strategy 4: Using Kitchen Logic
When a question asks the test taker to identify a main idea, you
should first focus on the opening and ending sentences of the passage
and each individual paragraph. If you can’t find the main idea from
these key sentences, then ask yourself how you would describe the
passage to someone who had never read it. Which words and phrases
would you use to explain the principle ideas of the passage?
This is called “Kitchen Logic” - when you explain something the way
you would if you were talking to your friends and family, while sitting
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at your kitchen table. So, when faced with identifying the main idea of
a difficult passage, make it easier on yourself by backing away from
the passage and thinking about it in terms of using easy “kitchen
logic”.

Strategy 5: Getting into the Author’s Mind
A number of questions become much easier when you place yourself
into the mind of the author of the passage. Ask yourself a few
different questions:

“Why did the author write this passage?”
“What was the author trying to say?”
What angle is the author taking?”
“What is the single most important point the author is trying to
make?”
Put yourself in the shoes of the author and imagine that you wrote the
passage and try to identify what you were trying to describe and how
you were trying to describe it. If you take on the opinions and ideas
expressed by the author as your own, then it becomes easier to
answer questions that would be easy for the author to answer.

Strategy 6: Emotional Words
Each question will be about a different angle of the passage. For
questions asking about the author’s emotions, find words in the
passage that are adjectives describing emotions.

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So, if a question asks what sort of attitude an author had towards the
passage or subject, then look throughout the passage for attitude
words that might convey a positive or negative attitude. Are words
such as brilliant, excited, delightful used, or are words such as
depressive, gloomy, disappointing used?
A lot of questions could be answered correctly simply by going through
and circling all the adjectives in a passage. Without looking at
anything else except for the adjectives in a passage, most questions
about attitude or emotion could be answered correctly.

Another way of handling these situations is to arrange all of the
answer choices in a list going from most negative to most positive.
Example:
Question: The author’s attitude on this topic is best described as:
A. indignation
B. eagerness
C. impartiality
D. fear
Now arrange these in order from negative to positive:
( - ) indignation, fear, impartiality, eagerness (+)
This will help sort out the different choices and keep you from
overlooking an answer choice and making an easy mistake.

Strategy 7: Finding the Key Words

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The strategy of finding certain “give-away” words does not only apply
to adjectives in questions about emotions or attitude. Many questions
about specific details will have key words that hold the “key” to finding
the right part of the passage to look in for the answer.
Rather than answering based on your memory of the passage, you
always want to have support for your answer choice rooted in a
specific part of the passage. To gain that support, it follows that you
have to identify which part of the passage to look in. While reading
back over the entire passage may be the most foolproof method of
finding that important part of the passage, it definitely is not the most

time economical method of finding that part of the passage.
A better route is to find key words in the question or answer choices
that are likely to stand out in the passage and will enable you to
quickly narrow your search down. These key words will be nouns or
verbs in the question or answer choices. Once you’ve identified
possible key words, then you should scan through the passage quickly
looking for either those key words to be repeated in the passage, or
their synonyms to appear in the passage. Once you find a particular
part of the passage that either has the exact key word repeated or a
synonym of the key word, you have probably identified the particular
part of the passage that will contain the support or justification that
you need to correctly answer the question and will allow you to be
confident in your answer choice selection.
One warning that should be made here is that often question writers
may use the exact same word or wording in their answer choices that
are used in the passage, but have done so in such a way as to mislead

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you. So, simply because a particular word or phrase appears in an
answer choice and also appears exactly the same in a passage does
not make that answer choice correct. Be sure that you reread the
answer choice and consider the context that it is in, to ensure that you
are not misled by a cheap trick.
In conclusion, always try to connect the question to the right words in
the passage that will allow you to save time in finding the right part of
the passage to look in for the answer and will give you the key to the

correct answer choice.

Strategy 8: Making Proper Inferences
Questions that ask you to make an inference from the passage will
require you to use your own personal judgment. Anything directly
stated by the author is not an inference. You will need to understand
the main idea of the passage in order to make a proper inference
about the author’s intent and mindset.
The obvious will not be enough to answer an inference question. You
must logically deduce what follows from what the author has stated in
the passage. You are looking for what can be inferred by the passage,
not what is directly stated in the passage.

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Strategy 9: Applying Ideas for Generalizations
Generalization questions are similar to inference questions in that you
have to go beyond what is directly stated in the passage by the
author. It helps to put yourself again in the author’s shoes. If you
were the author and believed in what you had just written, how would
you feel about another similar situation? What would either
strengthen or weaken your argument. How would you apply the
information you have just expressed to a completely different
situation?

Strategy 10: Using Context Clues
Context clues are a valuable aide in helping you understand difficult

phrases or words in the passage. A number of questions will ask you
about the meaning of words as they are used in a given passage.
If you already know the definition of the word, or have some
familiarity with it, a common mistake is to go with your first impulse
and choose the answer that you immediately recognize. However, the
reason the test writers may have chosen that particular vocabulary
word is because it is used in an unusual context. Therefore, return to
the passage and find where the word is used and make sure that you
understand how it is being used in the passage.
Once you’ve made your choice of a good definition go back again to
the passage and reread that particular section, but mentally replace
the answer choice you’ve chosen for the word being asked about.

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Example:
A passage states: “He was notorious for making decisions on the spur
of the moment…”
Question: Which of the following words, if substituted for the word
“notorious” would introduce the LEAST change in the meaning of the
sentence?
A. evil
B. disturbed
C. famous
D. despised
If you knew that the most common definition for “notorious” meant
being known in an unfavorable sense, then you might be tempted to

choose choice A, “evil.”
But once you review back over the passage, choice C, “famous” fits in
better into the context of the sentence of passage. Read the sentence
again and substitute your chosen answer choice for the word it
replaces. This gives you:
““He was famous for making decisions on the spur of the moment…,”
which makes sense and is correct.

Strategy 11: Breaking Down Passage Organization
In trying to understand the author’s perspective, you will sometimes
be asked about how the passage is organized. Many times, the
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simplest way to find the answer is to note how the opening sentence in
a passage or paragraph relates to the rest of the passage. How does
the author’s main idea get developed and broken down into supporting
ideas and statements?
As you go through the answer choices for these organization problems,
quiz yourself on each answer choice.
Example:
Question: Which of the following best describes the organization of the
author’s discussion of this topic?
A. He provides an example – Ask yourself, is there an example in
the question? Don’t work exclusively from your memory. Make
sure you can go back and actually find the example in the
passage.
B. He makes a comparison – Ask yourself, is there a comparison in

the question? Again, go back to the passage and actually find
the comparison being made and verify that it exists.
C. He makes an acknowledgement – Ask yourself, where is the
acknowledgement made and to whom?
D. He discusses a theory – Ask yourself, which theory is being
discussed?
After each of these initial questions, remember that it is not enough
for them simply to be true, they have to answer the question. Simply
because the author provided an example, doesn’t make choice A
correct. The example provided may have been to support a
comparison that he was making and the comparison may be the main
method of organization, which in this case would make answer choice

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B correct. So always read all the answer choices and only choose the
one that is the best, not just the first one you read that is factually
correct.

Strategy 12: First Word Analysis
When asked for main ideas that best summarize the passage, an easy
strategy is to look at the first words in each answer choice and without
looking at the rest of the answer choice, see if you could make a
decision based on those first words alone.
Example:
Question: Which of the following best explains the author’s primary
purpose?

A. dispute…
B. describe…
C. condemn…
D. convince…
If you know that the passage is fairly neutral about the subject, then
even if you know nothing else, you can probably eliminate the stronger
verbs used in answer choices A, C, and D, leaving you with “describe”
or answer choice B as being correct.

Strategy 13: Understanding the Intimidation
The test writers will generally choose passages that will be completely
foreign to most test takers. You can’t expect the passages to be on a
topic with which you have any familiarity. If you do happen to come
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across a passage that you are familiar with, consider yourself lucky,
but don’t plan on that happening.
The passages will also frequently be drawn from longer passages in
books, articles, journals, etc. Therefore, the passage that you will face
on the test may almost seem out of context and as though it begins in
the middle of a thought process. You won’t have a nice title overhead
explaining the general topic being covered but will immediately be
thrown into the middle of a strange format that you don’t recognize.
Also, while the topics chosen may have originally been interesting
reading in their original state, after a particular section is pulled and
used for the test passage, it will likely be dry and boring.
Getting hit by strange reading topics that you don’t recognize, of which

you may only have a small part of the original selection, and that are
dry and boring can be a bit intimidating if you’re not adequately
prepared. Just remember that the passages themselves will contain
all the information necessary to answer the questions and you don’t
need any prior knowledge of the topic in order to succeed and do well
on the test.

Strategy 14: Finding your Optimal Pace
Everyone reads at a different rate. It will take practice to determine
what is the optimal rate at which you can read fast and yet absorb and
comprehend the information. This is true for both the flyover that you
should initially conduct and then the subsequent reading you will have
to do as you go through and begin answering the questions. However,
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on the flyover, you are looking for only a surface level knowledge and
are not trying to comprehend the minutia of details that will be
contained in the passages.
You can practice with any form of reading material. Read an article at
your normal pace and then after you’re finished, ask yourself some
questions about what you just read and see how well you can
comprehend. Experiment with reading articles faster and slower and
always gauge how well you comprehended what you read at the end.
Train your brain to remember the details and absorb the facts.
With practice, you will find the pace that you should maintain on the
test while going back through passages. It should be a comfortable
rate. This is not a speed reading exercise. If you have a good pace,

and don’t spend too much time on any question, you should have a
sufficient amount of time to read the different sections of the passages
at a comfortable rate. The two extremes you want to avoid are the
dumbfounded mode, in which you are lip reading every word
individually and mouthing each word as though in a stupor, and the
overwhelmed mode, where you are panicked and are buzzing back and
forth through the passage in a frenzy and not comprehending
anything.
You must find your own pace that is relaxed and focused, allowing you
to have time for every question and give you optimal comprehension.
Note that you are looking for optimal comprehension, not maximum
comprehension. If you spent hours on each word and memorized the
passage, you would have maximum comprehension. That isn’t the
goal though, you want to optimize how much you comprehend with

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how much time you spend reading. Practice will allow you to
determine that optimal rate.

Strategy 15: Don’t be a Perfectionist
If you’re a perfectionist, this may be one of the hardest strategies, and
yet one of the most important. The test you are taking is timed, and
you cannot afford to spend too much time on any one question.
If you are working on a problem and you’ve got your answer split
between two possible answer choices, and you’re going back through
the passage and reading it over and over again in order to decide

between the two, you can be in one of the most frustrating situations
possible. You feel that if you just spent one more minute on the
problem, that you would be able to figure the right answer out and
decide between the two. Watch out! You can easily get so absorbed
in that problem that you loose track of time, get off track and end up
spending the rest of the test playing catch up because of all the
wasted time, which may leave you rattled and cause you to miss even
more questions that you would have otherwise.
Therefore, unless you will only be satisfied with a perfect score and
your abilities are in the top .1% strata of test takers, you should not
go into the test with the mindset that you’ve got to get every question
right. It is far better to accept that you will have to guess on some
questions and possibly get them wrong and still have time for every
question, than to work on every problem until you’re absolutely
confident in your answer and then run out of time on the last few
problems.
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Strategy 16: Factually Correct, but Actually Wrong
A favorite ploy of question writers is to write answer choices that are
factually correct on their own, but fail to answer the question, and so
are actually wrong.
When you are going through the answer choices and one jumps out for
being factually correct, watch out. Before you mark it as your answer
choice, first make sure that you go back to the question and confirm
that the answer choice answers the question being asked.

Strategy 17: Different Viewpoints
Some passages will express the author’s viewpoint on a topic, along
with the viewpoint of other experts or other individuals. This can lead
to trouble in answering questions though. If asked for the viewpoint of
the author, you might go back to the passage, find where a certain
viewpoint is expressed, answer the question based on what you read
and move on.
For most passages, that would be fine, but when other viewpoints
besides the author’s are expressed, you have to discern who is
expressing their opinion in the passage. Make sure that if multiple
individuals are giving their viewpoint on a topic, that you sort them out
for any questions and associate the right viewpoint with the right
individual.

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Strategy 18: Extraneous Information
Some answer choices will seem to fit in and answer the question being
asked. They might even be factually correct. Everything seems to
check out, so what could possibly be wrong?
Does the answer choice actually match the passage, or is it based on
extraneous information not even contained in the passage. Just
because an answer choice seems right, don’t assume that you
overlooked information while reading the passage. Always try to go
back and find the support for the answer choice in the passage. Your
mind can easily play tricks on you and make you think that you read
something or that you overlooked a phrase.

Unless you are behind on time, always go back to the passage and
make sure that the answer choice “checks out.”

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Mathematics
The ACT Mathematics test measures a test taker’s ability to solve
problems representing some of the key concepts in mathematics.
Some problems will only test one concept, while others will involve
multiple concepts integrated together in a single problem.
The problems will have few technical terms, aside from basics, such as
area, perimeter, integer, and ratio, which are expected to be common
mathematical knowledge. All figures shown will be drawn accurately
and lie in a single plane, unless noted otherwise.

Number Types
Integers, Odd and Even Numbers, Prime Numbers, Digits

ƒ

Integers…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …

ƒ

Consecutive Integers: Integers that follow in sequence; for
example, 22, 23, 24, 25. Consecutive Integers can be more
generally represented by n, n + 1, n + 2, n + 3, …

ƒ

Odd Numbers…, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, …

ƒ

Even Numbers…, -8, -6, -4, -2, 0, 2, 4, 6, 8, … (Note: zero is
an even number)

ƒ

Prime Numbers…, 2, 3, 5, 7, 11, 13, 17, 19, … (Note 1 is not a
prime and 2 is the only even prime)

ƒ

Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Addition and Multiplication of Odd and Even Numbers

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Addition

Multiplication

even + even = even

even x even = even

odd + odd = even

even x odd = even

even + odd = odd

odd x odd = odd

Percent
Percent means hundredths or number out of 100. For example, 40
percent means 40/100 or .40 or 2/5.
Percent less than 100

Problem 1: If the sales tax on a $30 item is $1.80, what is the sales
tax rate?
Solution: $1.80 = n/100 x $30
n = 6, so 6% is the sale tax rate
Percent Greater than 100

Problem 2: What number is 250% of 2?
Solution: n = 250/100 x 2
n = 5, so 5 is the number
Percent less than 1

Problem 3: 3 is 0.2 percent of what number?
Solution: 3 = 0.2/100 x n

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n = 1,500, so 1,500 is the number
Percent Increase/Decrease

Problem 4: If the price of a computer was decreased from $1,000 to
$750, by what percent was the price decreased?
Solution: The price decrease is $250. The percent decrease is the
value of n in the equation 250/1000 = n/100. The value of n is 25, so
the price was decreased by 25%.
Notes: n% increase means increase/original = n/100;
n% decrease means decrease/original = n/100.

Average
An average is a statistic that is used to summarize data. The most
common type of average is the arithmetic mean. The average
(arithmetic mean) of a list of n numbers is equal to the sum of the
numbers divided by n. For example, the mean of 2, 3, 5, 7, and 13 is
equal to
2 + 3 + 5 + 7 + 13 / 5 = 6
When the average of a list of n numbers is given, the sum of the
numbers can be found. For example if the average of six numbers is
12, the sum of these six numbers is 12 x 6, or 72.

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The median of a list of numbers is the number in the middle when the
numbers are ordered from greatest to least or from least to greatest.
For example, the median of 3, 8, 2, 6, and 9 is 6 because when the
numbers are ordered, 2, 3, 6, 8, 9, the number in the middle is 6.
When there is an even number of values, the median is the same as
the mean of the two middle numbers. For example, the median of 6,
8, 9, 13, 14, and 16 is
9 + 13 / 2 = 11
The mode of a list of numbers is the number that occurs most often in
the list.

For example, 7 is the mode of 2, 7, 5, 8, 7, and 12.

The

numbers 10, 12, 14, 16, and 18 have no mode and the numbers 2, 4,
2, 8, 2, 4, 7, 4, 9, and 11 have two modes, 2 and 4.
Note: The mean, median, and mode can each be considered an
average.

On the test, the use of the word average refers the

arithmetic mean and is indicated by “average (arithmetic mean).” The
exception is when a question involves average speed (see problem 2
below).

Questions involving the median and mode will have those

terms stated as part of the question’s text.

Weighted Average
Problem 1: In a group of 10 students, 7 are 13 years old and 3 are
17 years old. What is the average (arithmetic mean) age of these 10
students?

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Solution: The solution is not the average of 13 and 17, which is 15.
In this case the average is
7(13) + 3(17) / 10 = 91 + 51 / 10 = 14.2 years
The expression “weighted average” comes from the fact that 13 gets a
weight factor of 7, whereas 17 gets a weight factor of 3.
Average Speed

Problem 2: Jane traveled for 2 hours at a rate of 70 kilometers per
hour and for 5 hours at a rate of 60 kilometers per hour. What was
her average speed for the 7-hour time period?
Solution: In this situation, the average speed is:
Total Distance/Total Time
The total distance is 2(70) + 5(60) = 440 km.
The total time is 7 hours. Thus the average speed was
440/7 = 62 6/7 kilometers per hour.
Note: In this example the average speed is not the average of the two
separate speeds, which would be 65.

Properties of Signed Numbers
positive x negative = negative

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